Recently, N. Masmoudi [1]} is concerning with the long time behavior for polymeric models. The co-rotation compressible FENE system has been studied in [2]}. In the co-rotation case, the drag term \(\sigma (u)=\frac{\nabla u-\nabla u^T}{2}\)  which leads to some good structure such that the \(\psi -\psi _\infty \)  is exponential decay in time. To our best knowledge, the same problem for the general case has not been studied yet. This problem is interesting and more difficult than the co-rotation case. In this paper, we firstly study the global well-posedness results for (REF ). The key point is to prove a global priori estimate for (REF ) with small data. Using the energy methods and the cancellation relation between the CNS equations and Fokker-Planck equation, for \(d\ge 2\) , we obtain a global priori estimate.
Moreover, if \(d\ge 3\) , we study about the large time behaviour and obtain the optimal time decay rate for \((\rho ,u)\)  in \(L^2\) . The proof is based on the Fourier splitting method and the Littlewood-Paley decomposition theory. The first difficult is to estimate the additional linear term \(div~\tau \) . Motivated by [3]} and [4]}, we can cancel the stress term \(\tau \)  in Fourier space. Then we obtain the time decay rate \((1+t)^{-\frac{d}{8}}\)  for the velocity in \(L^2\)  by the Fourier splitting method and the bootstrap argument. The main difficult to get optimal time decay rate is that we can not get any information of \(u\)  in \(L^1\)  from (REF ). Fortunately, similar to [5]}, we can prove a slightly weaker conclusion \(\Vert u\Vert _{L^\infty (0,\infty ; \dot{B}^{-\frac{d}{2}}_{2,\infty })}\le C\)  from (REF ) by using the time decay rate \((1+t)^{-\frac{d}{8}}\) . Finally, we obtain optimal time decay rate for the velocity in \(L^2\)  by the Littlewood-Paley decomposition theory and the standard Fourier splitting method.
